Theorem resco 5051

Description: Associative law for the restriction of a composition. (Contributed by NM, 12-Dec-2006.)

Assertion

Ref	Expression
resco	|- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )

Proof of Theorem resco

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relres 4855	. 2 |- Rel ( ( A o. B ) |` C )
2		relco 5045	. 2 |- Rel ( A o. ( B |` C ) )
3		vex 2692	. . . . . 6 |- x e. _V
4		vex 2692	. . . . . 6 |- y e. _V
5	3, 4	brco 4718	. . . . 5 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
6	5	anbi1i 454	. . . 4 |- ( ( x ( A o. B ) y /\ x e. C ) <-> ( E. z ( x B z /\ z A y ) /\ x e. C ) )
7		19.41v 1875	. . . 4 |- ( E. z ( ( x B z /\ z A y ) /\ x e. C ) <-> ( E. z ( x B z /\ z A y ) /\ x e. C ) )
8		an32 552	. . . . . 6 |- ( ( ( x B z /\ z A y ) /\ x e. C ) <-> ( ( x B z /\ x e. C ) /\ z A y ) )
9		vex 2692	. . . . . . . 8 |- z e. _V
10	9	brres 4833	. . . . . . 7 |- ( x ( B |` C ) z <-> ( x B z /\ x e. C ) )
11	10	anbi1i 454	. . . . . 6 |- ( ( x ( B |` C ) z /\ z A y ) <-> ( ( x B z /\ x e. C ) /\ z A y ) )
12	8, 11	bitr4i 186	. . . . 5 |- ( ( ( x B z /\ z A y ) /\ x e. C ) <-> ( x ( B |` C ) z /\ z A y ) )
13	12	exbii 1585	. . . 4 |- ( E. z ( ( x B z /\ z A y ) /\ x e. C ) <-> E. z ( x ( B |` C ) z /\ z A y ) )
14	6, 7, 13	3bitr2i 207	. . 3 |- ( ( x ( A o. B ) y /\ x e. C ) <-> E. z ( x ( B |` C ) z /\ z A y ) )
15	4	brres 4833	. . 3 |- ( x ( ( A o. B ) |` C ) y <-> ( x ( A o. B ) y /\ x e. C ) )
16	3, 4	brco 4718	. . 3 |- ( x ( A o. ( B |` C ) ) y <-> E. z ( x ( B |` C ) z /\ z A y ) )
17	14, 15, 16	3bitr4i 211	. 2 |- ( x ( ( A o. B ) |` C ) y <-> x ( A o. ( B |` C ) ) y )
18	1, 2, 17	eqbrriv 4642	1 |- ( ( A o. B ) |` C ) = ( A o. ( B |` C ) )

Syntax hints:   /\ wa 103   = wceq 1332   E.wex 1469   e. wcel 1481   class class class wbr 3937   |` cres 4549   o. ccom 4551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-co 4556  df-res 4559

This theorem is referenced by:  cocnvcnv2  5058  coires1  5064  relcoi1  5078  dftpos2  6166